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Unformatted text preview: Independence Irrelevant information Based on a new information, we can update the sample space and events to be new ones, so that we can calculate a conditional probability of the updated event. However, sometimes, the new information may be irrelevant. Therefore, the conditional probability of B given A, P(B|A), is exactly the same as the unconditional probability of B. That is, we do not obtain any new relevant information from the event A to update the probability of B. In such a case, we would say that B and A are (statistically) independent . If two events are NOT independent, then we would say that they are dependent. Independence ) ( ) ( ) ( B P A P B A P = Definition (Independence) ) ( ) | ( B P A B P = ) ( A P when Two events A and B are called (statistically) independent if and ony if OR, equivalently, Theorem: If events A and B are independent, then all of A and B C , A C and B , and A C and B C are also independent. Independence Prove it later. Example On page 25 of the course note Question 14: A fair coin is tossed three times. Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , show that 1) Events A and B are independent; 2) Events B and C are dependent. Example 8 3 ) ( , 2 1 ) ( , 4 1 ) ( = = = C P B P A P On page 25 of the course note Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , Events A and B are independent? ? ) ( = B A P Example 8 3 ) ( , 2 1 ) ( , 4 1 ) ( = = = C P B P A P On page 25 of the course note Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , Events A and B are independent? 8 1 ) ( = B A P =P( A ) X P( B ) Example 8 3 ) ( , 2 1 ) ( , 4 1 ) ( = = = C P B P A P On page 25 of the course note Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , Events B and C are independent? ? ) ( = B C P Example 8 3 ) ( , 2 1 ) ( , 4 1 ) ( = = = C P B P A P On page 25 of the course note Denote the event that a head occurs on each of the first two tosses by A , the event that a tail occurs on the third toss by B , and the event that exactly two tails occur in the three tosses by C , Events B and C are independent?...
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